Final Fantasy IV Side Quest - Cave of Summon Monsters
This sidequest is important if you want to get summons such as Bahamut, Odin, Leviathan and Asura. Head to the underground world (the Lunar Whale can't fit, so take the Falcon), then so southwest until you see a small isle with a cavern entrance, land your ship and enter. Rydia will tell you that she came from there, so they should ask for help. Do not step on the flashing tiles, since they will decrease the party's HP by 40. So, avoid them when you can, but when you can't, cast Float on the party. Head southeast to the treasure chest and get a Cure2 from inside. Go all the way north and open the treasure chest for a Life. Head west into the secret passageway and get the treasure chest there for Ether1. Go back east, then south, and northwest and to the next floor by the stairs.

Now, go east, then southeast and open the treasure chest there for a Cure2. Go up a little, then northwest. Open the treasure chest for a Cabin. Go east and down the stairs. Follow the pathway until you reach a treasure chest. Open it for another Life. Go all the way east, and open the treasure chest for a Cure3. Go east, follow the secret pasageway south and go east to the treasure chests. They contain a Cure2, a Poison axe and a Ninja sword. Make your way back to the secret passageway. Head northwest and open the treasure chest for a monster battle. Defeat the monsters and get the Defense Sword from within. Now go back to the teleporter in the center of the room. You'll end up in the populated area of the summon monsters cave.

Go northwest as you enter and open the treasure chest for an Ether1. Go north up the stairs and open the treasure chest for 2000GP. Go all the way east and open the next treasure chest you see for another 2000GP. Go south and open the treasure chest for 3000GP. Step back on that teleporter you saw earlier up north. When you warp to the next floor, warp back and open the previous unaccessable treasure chest for a Rat's Tail. This is an extremely important item. Go back on the teleporter. You'll be in the town of summon monsters now. There are a variety of shops to look in, just like normal shops in towns. You have to head to the library west of the town, next to the chocobo. If you want, rest at the inn and buy some items at the item shop. Once you're finished, head to the library. Head down the stairs and onto the teleporter. You'll be in the lair of King Leviathan and Queen Asura. Speak to Queen Asura, say Yes when prompted and prepare for a tough battle.

Queen AsuraHP: 31005EXP: 20000GP: 0

Queen Asura isn't very difficult per se, but it's impossible to win the battle without casting Wall on her first. She has moderately high HP but doesn't
attack the party too often (even when she does, the most I've seen is somewhere between 400-500HP). She'll Cure herself almost non-stop in the battle, which is why you need to cast Wall on her so her healing spells can bounce back on your party. Then, have Cecil attack, Kain jump, Edge throw, Rydia use Titan and Rosa should use her arrows, since Asura will heal the party in place of Rosa.

After you win the battle with Asura, you can use her as a summon; use a tent, save then come back and talk to the King, standing next to Asura (although he previously said he was not the King). Select Yes when prompted and prepare to battle Leviathan.

LeviathanHP: 50001EXP: 28000GP: 0

Lightning is your key to winning this battle. Leviathan will start out using Big Wave, which will do a lot of damage to your party, so have Rydia use her new Asura summon and Lit3 spell, Rosa should Cure the party while Kain Jumps, Cecil attacks and Edge uses Blitz and Throw. Leviathan may use weak Ice spells, but they don't do so much damage. It's not such a hard battle.

Leviathan will join you as a summon afterwards. Congratulations on completing the Cave of Summon Monsters sidequest! You can now get the Odin and Bahamut. Use Rosa's Exit spell to wrap back up out of the cave.